Well my method was to go to school for a year, then work for a year off and on, taking general classes at community college since that was massively cheaper, then transferring them over to a university later to finish off my degree. Point is, I think you can understand the more barriers there are to education, the longer the next stage of human evolution as you call it is going to take.You know you can still enroll in the military to get free college just like baby boomers. Worked just fine for my dad, my brother, and my sister.
There are a lot of ways to address this. Don't want to work and be a slackass? No problem, you get to live in projects-grade housing and eat the gourmet equivalent of dog food. Enjoy life! Oh, you would like to live in a HOUSE? Yeah, you'll have to work for that. Or hey, if daddy made millions as a businessman, well that was HIM. You could put caps on how much wealth could be transferred to his descendants to prevent modern-day aristocracies.Not at all, because you can't teach an old dog new tricks. By the time they are rich, they have generally crystallized their world view, attitude, and personality, and aren't going to suddenly change into different people when they hit it big. They are going to keep acting how they have always acted.
The problem is, if you have children that see mommy and daddy just sitting on their asses watching TV and taking that government check, how do you keep them from mimicking what they see around them and acting the same? If that's the natural state of man, and there's no carrot to drive people, its likely to be common, and its also likely that these people are going to have far more children than the successful career oriented driven people.
In other words, how do you prevent Idiocracy or Wall-E? I don't have an answer either.
People can and will strive to better their condition, that's as natural as eating. Like you said though, there has to be a carrot to think that they CAN in the first place as opposed to being simply fucked. I'm saying homelessness and hunger are bad "sticks" for trying to advance humanity if it can at all be avoided. What's the point of civilization if you're not any further from death in society than you would be in the jungle?